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When it comes to multiplying binomials, the FOIL method is a straightforward approach that many students find useful. FOIL is an acronym representing the steps in the process: **First, Outer, Inner, Last**. Each step refers to the terms in the binomials that are multiplied together.

Let's break it down using the expression \((x + \frac{1}{8})(x - \frac{9}{8})\):

• First: Multiply the first terms in each binomial: \(x \times x = x^2\).
• Outer: Multiply the outer terms: \(x \times -\frac{9}{8} = -\frac{9}{8}x\).
• Inner: Multiply the inner terms: \(\frac{1}{8} \times x = \frac{1}{8}x\).
• Last: Multiply the last terms: \(\frac{1}{8} \times -\frac{9}{8} = -\frac{9}{64}\).

Once all these steps are completed, you're left with a list of terms that can then be simplified into a single expression.

After applying the FOIL method, we end up with several terms. To make the expression simpler, we must combine like terms. Like terms are terms that have the same variable raised to the same power.

From the expression \(x^2 -\frac{9}{8}x + \frac{1}{8}x - \frac{9}{64}\), the terms \(-\frac{9}{8}x\) and \(\frac{1}{8}x\) can be combined, because they both contain the variable \(x\) raised to the first power.

By adding these two terms together, \(-\frac{9}{8}x + \frac{1}{8}x\), we obtain \(-1x\) or simply \(-x\). This step reduces the complexity of the expression, making it easier to understand and work with.

The final stage in solving the problem is to simplify the polynomial expression. Polynomial simplification is the process of combining like terms and ensuring the expression is presented in its simplest form.

Starting with \(x^2 -\frac{9}{8}x + \frac{1}{8}x - \frac{9}{64}\), after combining the like terms of \(-\frac{9}{8}x + \frac{1}{8}x\) into \(-x\), we are left with the polynomial: \[x^2 - x - \frac{9}{64}.\]

This expression is now fully simplified and ready to be used in solving other parts of a problem or in further operations such as graphing or factoring. Simplifying polynomials helps reveal the true nature of the expression, ensuring that it is as concise and easy to work with as possible.